A307318 - OEIS

%I #16 May 21 2019 03:49:36

%S 1,-2,37,-692,14371,-315002,7156969,-166785320,3960790687,

%T -95442311582,2326713829837,-57260397539204,1420295354815351,

%U -35463581316556850,890530353765972817,-22472131364683145552,569507678494598796631,-14487492070374441746150

%N a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

%H Seiichi Manyama, <a href="/A307318/b307318.txt">Table of n, a(n) for n = 0..100</a>

%F From _Vaclav Kotesovec_, Apr 02 2019: (Start)

%F Recurrence: 6*(n-1)*n^2*(490*n^4 - 3948*n^3 + 11668*n^2 - 14967*n + 7027)*a(n) =  - (n-1)*(74480*n^6 - 675066*n^5 + 2399756*n^4 - 4233492*n^3 + 3852029*n^2 - 1682577*n + 272160)*a(n-1) + (131320*n^7 - 1437814*n^6 + 6472114*n^5 - 15414556*n^4 + 20770423*n^3 - 15610855*n^2 + 5939868*n - 861840)*a(n-2) - (27440*n^7 - 355838*n^6 + 1853810*n^5 - 4998800*n^4 + 7460459*n^3 - 6071312*n^2 + 2439561*n - 362880)*a(n-3) - 3*(2*n - 5)*(3*n - 8)*(3*n - 7)*(490*n^4 - 1988*n^3 + 2764*n^2 - 1515*n + 270)*a(n-4).

%F a(n) ~ (-1)^n * 3^(3*n + 7/2) / (128*Pi*n). (End)

%t Table[Sum[(-1)^(i + j + k) * (i + j + k)!/(i!*j!*k!), {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 02 2019 *)

%o (PARI) {a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}

%o (PARI) {a(n) = sum(i=0, 3*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^3, i))} \\ _Seiichi Manyama_, May 20 2019

%Y Cf. A120305, A144660, A307324.

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 02 2019